**TASHKENT UNIVERSITY OF INFORMATION TECHNOLOGIES NAMED AFTER MUHAMMAD AL-KHORAZMI, MINISTRY OF INFORMATION TECHNOLOGY AND COMMUNICATION DEVELOPMENT OF THE REPUBLIC OF UZBEKISTAN**
__Discrete structures __
**Topic : Euler and Gamiltonian cycles and graphs**
Group: MTH401 Student: Yuldoshov Dilshodbek Teacher: Sabirov Karimjon
**Plan:**
**1.Cycle of Euler Hamilton**
**2. Euler graphs**
**3.Hamiltonian graphs Euler graphs**
**Euler**
Euler Leonard [1707.4(15)4, Basel, Switzerland — 1783.7(18).9, Petersburg] — mathematician, mechanic and physicist, member of Petersburg FA. President of the Berlin FA (since 1759). In 1720-24, he studied at the University of Basel (Switzerland) and listened to I. Bernoulli's lectures on mathematics. He worked at St. Petersburg FA and worked at Berlin FA without severing academic ties with FA. His main scientific works are related to all branches of mathematics of that time, mechanics, theory of elasticity, mathematical physics, optics, theory of music, theory of machines, ballistics, marine science and others. Euler's discoveries in the mechanics of the heavens, the mechanics of the surrounding medium, and optics occupy an important place. Euler founded areas such as the calculus of variations, number theory, the theory of analytic and special functions, the theory of differential equations, and the theory of surfaces. Euler's angles, introduced by him, occupy an important place in mechanics. Series scientific works are related to analytic geometry, series theory and topology (see Euler's formulas).
In honor of L. Euler, the cycle containing all the edges of the graph was called the Euler line, the Euler cycle, the closed Euler chain, or simply the Euler chain.
The graph containing the Euler cycle is called the Euler graph.
If a graph contains an open circuit containing all the edges of this graph, then such a graph is called semi-Euler
*Here are some of the most important theorems about Euler graphs.*
Theorem #1
Theorem #1. If all vertices in a connected graph are even, then this graph contains an Euler cycle.
Example: in Figure shows a graph in which the degrees of all vertices are even. You can start traversing its edges from any vertex. Let us denote the edges by the letters a, b, c, d, e, f, k, m, n. Then the following sequence of edges and vertices can serve as an example of an Euler cycle:
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